Quasilinear Diffusion Equation Research Articles

The effect of altitude- and velocity-dependent wave–particle interactions on the H + and O + outflows in the auroral region

The formation of H + and O + toroids at high altitudes in the auroral region are discussed in terms of altitude- and velocity-dependent wave–particle interactions (WPI), and special attention is given to the peaked nature of the velocity diffusion coefficient D ⊥. The effects of altitude- and velocity-dependent WPI are taken into account by perturbing the ion velocity with random increment Δ v ⊥, such that 〈 ( Δ v ⊥ ) 2 〉 = 4 D ⊥ Δ t , where the time step is Δ t. To model the heating process, we specify a model for the velocity diffusion coefficient as a function of the ion perpendicular velocity and position along the auroral geomagnetic field line. The ion velocity distribution is described by a quasi-linear diffusion equation, which is solved by the Monte Carlo technique. The Monte Carlo model includes the effects of altitude- and velocity-dependent WPI, gravity, polarization electrostatic field, and the divergence of the geomagnetic field within the simulation tube (1.2–10 earth radii, R E). These effects were included self-consistently in the computations. The peaked nature of D ⊥ reflects the way in which altitude- and velocity-dependent WPI lead to the formation of H + and O + toroids at high altitudes in the auroral region, rather than simple bulk heating process. Because D ⊥ falls to zero at small perpendicular velocities, the bulk of the ion velocity distribution is unaffected by interaction with the waves (electromagnetic turbulence). However, near the phase velocity v 0, D ⊥ begins to become appreciable, and the diffusion process begins to affect ions. Because there are initially more ions at lower velocities than at higher velocities, the net escape flux in velocity space is toward higher velocities, leading to the formation of ion toroids. At large perpendicular velocities, D ⊥ falls to zero and consequently, the effect of WPI is negligible. As the heated ions drift upward along geomagnetic field lines due to the mirror geometry of the Earth's magnetic field, they eventually leave the primary heating region and form a ring “donuts”. The heating process is found to be self-limiting, and this explains the saturation of the ion velocity distributions at high altitudes. The altitude profiles of ion density, drift velocity, parallel and perpendicular temperatures are also discussed. We find that including the effect of velocity-dependent WPI in addition to the effect of altitude-dependent WPI produce realistic ion temperatures that are, qualitatively, comparable to the observations. The model produces simulation results similar to the observed toroids.

Nonlinear theory of cyclotron resonant wave-particle interactions: Analytical results beyond the quasilinear approximation

Cyclotron resonant wave-particle interactions are studied in the context of Hamiltonian theory with utilization of Lie transform techniques. The canonical perturbation method for single particle motion is used for providing results for the collective particle behavior under interaction with wave fields of either localized or periodic profiles. Analytical expressions for the calculation of phase-averaged quantities of physical interest as well as the diffusion equation are derived. In the lowest order of perturbation, the method reformulates in a rigorous and unifying context the derivation of well-known results, namely Madey's theorem and quasilinear diffusion equation. Proceeding to higher order the method provides results consisting of fourth-order accurate analytical expressions for the calculation of phase-averaged quantities as well as the derivation of a fourth-order accurate diffusion equation, with higher-order derivatives, which is the extension of the well-known Fokker-Planck equation beyond the quasilinear approximation. Higher-order terms are related to the effect of nonlinear resonant coupling between different spectral components of the waves, on the evolution of the particle distribution function.

On a class of equations with variable parabolicity direction

We consider the following pseudoparabolic regularization of a forward-backward quasilinear diffusion equation: $u_t=\Delta \phi(u)+\varepsilon\Delta u_t$ ($\varepsilon>0$). As suggested by several models of the applied sciences, the function $\phi$ is nonmonotone and vanishing at infinity. We investigate the limit points of the set of solutions to the associated Neumann problem as $\varepsilon\to 0$, proving existence of suitable weak solutions of the original ill-posed equation. Qualitative properties of such solutions are also addressed.

Spin-polarized transport: Existence of weak solutions

A system modeling spin-polarized transport in ferromagneticmultilayers is considered. In this model, the spin accumulation isdescribed by a quasilinear diffusion equation with discontinuous,measurable coefficients. This equation is coupled to theLandau-Lifshitz-Gilbert equation, a nonlinear, nonlocal equationdescribing the precession of themagnetization in the ferromagnetic layers. The global existence ofweak solutions is proved.

Distortion-suppression optimization for a class of nonlinear optical systems with feedback

We consider the initial boundary-value problem for the quasilinear diffusion equation ∂ t u+u − DΔu = K(x, y)(1 + γ(u + ϕ(x, y))) describing the dynamics of optical systems with controlled feedback wave intensity modulation K(x, y) in the presence of incoming-wave phase perturbations ϕ(x, y). The control problem for the parameter K(x, y) is formulated with the objective of smoothing out the spatial nonhomogeneities of the total output phase u(x, y, T) + ϕ(x, y). We prove existence and uniqueness theorems for the generalized solutions of the direct and conjugate problems, solvability theorems for the optimization problems, and Frechet-differentiability of the objective functional. A formula for the functional gradient is derived and the efficiency of the gradient projection method is demonstrated numerically.

Relativistic quasilinear diffusion in axisymmetric magnetic geometry for arbitrary-frequency electromagnetic fluctuations

A relativistic bounce-averaged quasilinear diffusion equation is derived to describe stochastic particle transport associated with arbitrary-frequency electromagnetic fluctuations in a nonuniform magnetized plasma. Expressions for the elements of a relativistic quasilinear diffusion tensor are calculated explicitly for magnetically trapped particle distributions in axisymmetric magnetic geometry in terms of gyro-drift-bounce wave-particle resonances. The resonances can destroy any one of the three invariants of the unperturbed guiding-center Hamiltonian dynamics.

Intermediate Asymptotics for Inhomogeneous Nonlinear Heat Conduction

We consider the initial-value problem for a quasilinear heat-conduction or diffusion equation with variable density decreasing at infinity. We show that the asymptotic behavior of the given process is self-similar. Indeed, as t → ∞ the solution of the problem approaches a self-similar solution of a certain singular “limit” equation. The limit solution has compact support for any t > 0 and cusp-type shape at the space origin.

Determination of a nonlinear source term in a diffusion equationwith final observations

This paper deals with an inverse problem of determining a nonlinear source term in a quasilinear diffusion equation with overposed final observations. Applying integral identity methods, data compatibilities are deduced by which the inverse source problem here is proved to be reasonable and solvable. Furthermore, with the aid of an integral identity that connects the unknown source terms with the known data, a conditional stability is established.

Bounded imaginary powers of differential operators on manifolds with conical singularities

We study the minimal and maximal closed extension of a differential operator A on a manifold B with conical singularities, when A acts as an unbounded operator on weighted L^p-spaces over B, 1 < p < \infty. Under suitable ellipticity assumptions we can define a family of complex powers A^z. We also obtain sufficient information on the resolvent of A to show the boundedness of the purely imaginary powers. Examples concern unique solvability and maximal regularity for the solution of the Cauchy problem for the Laplacian on conical manifolds as well as certain quasilinear diffusion equations.

On resonant interactions of ions with plasma waves in a reduced quasi-linear theory

Abstract. Based on quasi-linear theory (involving pitch angle scattering), the resonant interactions between ions and waves in an anisotropic multi-component plasma are discussed. In particular, electromagnetic Alfvén and ion-cyclotron waves propagating along or obliquely to the magnetic field are considered. A set of reduced (with respect to the perpendicular velocity component) quasi-linear diffusion equations is derived, involving reduced 1-D velocity distribution functions (VDFs), as they occur in wave dispersion relations. A 2-D model VDF can be constructed when using the Gaussian approximation. Wave-particle heating and acceleration rates are calculated.

Relativistic bounce-averaged quasilinear diffusion equation for low-frequency electromagnetic fluctuations

A relativistic bounce-averaged quasilinear diffusion equation is derived to describe stochastic particle transport associated with low-frequency electromagnetic fluctuations in a nonuniform magnetized plasma. Expressions for the relativistic quasilinear diffusion coefficients are calculated explicitly for magnetically-trapped particle distributions in axisymmetric magnetic geometry in terms of drift-bounce resonant contributions associated with low-frequency fluctuations which conserve the first adiabatic invariant.

Generalized weak turbulence theory

In this article we present the derivation of a generalized weak turbulence kinetic equation for unmagnetized collisionless plasmas in a uniform medium. For the sake of simplicity the present formulation assumes longitudinal electrostatic interaction only, and the effects of spontaneous thermal fluctuations are ignored. In spite of these simplifications, the present formalism represents a generalization of the existing weak turbulence theory in that a nonlinear eigenmode excited in a turbulent plasma with frequency close to twice the plasma frequency is incorporated into the discussion. Traditional weak turbulence theory emphasizes various linear and nonlinear interactions among wave modes in quiescent plasmas (i.e., Langmuir and ion-sound waves). In contrast, the present formalism describes linear and nonlinear interactions among Langmuir, ion-sound, and the new nonlinear eigenmode. Nonlinear wave kinetic equations for these modes are systematically derived, and the particle kinetic equation which generalizes the well known quasilinear diffusion equation, is also derived.

Separation of variables and exact solutions to quasilinear diffusion equations with nonlinear source

A solution of a partial differential equation with two real variables t and x is functionally separable in these variables if q( u)= φ( x)+ ψ( t) for some single variable functions, q, φ and ψ. In this paper, the generalized conditional symmetry approach is used to study the separation of variables of quasilinear diffusion equations with nonlinear source. We obtain a complete list of canonical forms for such equations which admit the functionally separable solutions. As a result, we get broad families of exact solutions to some quasilinear diffusion equations with nonlinear source. The behavior and blow-up properties of some solutions are described.

Quantitative comparison of reduced-description particle-in-cell and quasilinear-Zakharov models for parametrically excited Langmuir turbulence

The effect of kinetic processes on the saturation of parametric instabilities in an electromagnetically driven plasma is investigated. A reduced-description particle-in-cell technique is used as a benchmark to test a new quasilinear-Zakharov model which accounts for electron heating due to Landau damping by coupling the quasilinear diffusion equation to the Zakharov equations. The reduced-description particle-in-cell method utilizes a two-time-scale approximation which significantly reduces the numerical dissipation and ion noise levels. This approach allows accurate modeling of Langmuir and ion acoustic waves in regimes typically studied with Zakharov simulations. The comparison of the two models is performed for the test case of a one-dimensional homogeneous plasma driven by a spatially uniform pump in both the Langmuir decay instability cascade and collapse regimes. Good agreement is found in both weakly and strongly driven regimes for the total Langmuir wave energy and evolved electron velocity distributions. Electron heating significantly decreases saturation levels in strongly driven regimes by increasing the Landau damping rate, bringing the quasilinear-Zakharov simulations in much closer agreement with the reduced-description particle-in-cell simulations than standard Zakharov simulations, which do not account for the evolution of the electron distribution.

Pitch-angle diffusion of relativistic electrons due to resonant interactions with whistler waves

Pitch-angle diffusion phenomenon caused by resonant interactions between parallel whistler waves and relativistic electrons is examined for the Van Allen radiation belts. Thus, the quasilinear pitch-angle diffusion equation for the relativistic case is derived. From this equation, in stationary conditions, one computes the wave spectrum, the distribution function, the particle lifetime, and the trapped electron flux between the magnetic mirrors. The present results may explain phenomena occurring in Earth’s magnetosphere.

Discussion

We consider a new formulation of the Fourier filtering problem based on the use of matrix filters instead of conventional filters-multiplicators in the framework of the periodic boundary value problem for a quasilinear diffusion equation of nonlinear optics. Our attention is mostly focused on studying the capabilities of matrix Fourier filtering as an efficient tool for constructing periodic solutions with prescribed spatial-temporal dynamics. To this end, we analyze the spectrum of the linearized operator and the possibility to localize it by choosing a matrix Fourier filter that provides the existence of the Hopf bifurcation. We also describe the structure of bifurcation solution for two important classes of filters: finite filters-multiplicators and matrix filters. We theoretically and numerically show that along with rotating waves, typical for the Hopf bifurcation with conventional filters-multiplicators, there exist waves with new nontrivial dynamics (rotating waves with nodes and pulsating structures). These waves excite as a result of the Hopf bifurcation under an appropriate choice of nondiagonal matrix Fourier filters.

Closure of multi-fluid and kinetic equations for cyclotron-resonant interactions of solar wind ions with Alfvén waves

Abstract. Based on quasilinear theory, a closure scheme for anisotropic multi-component fluid equations is developed for the wave-particle interactions of ions with electromagnetic Alfvén and ion-cyclotron waves propagating along the mean magnetic field. Acceleration and heating rates are calculated. They may be used in the multi-fluid momentum and energy equations as anomalous transport terms. The corresponding evolution equation for the average wave spectrum is established, and the effective growth/damping rate for the spectrum is calculated. Given a simple power-law spectrum, an anomalous collision frequency can be derived which depends on the slope and average intensity of the spectrum, and on the gyrofrequency and the differential motion (with respect to the wave frame) of the actual ion species considered. The wave-particle interaction terms attain simple forms resembling the ones for collisional friction and temperature anisotropy relaxation (due to pitch angle scattering) with collision rates that are proportional to the gyrofrequency but diminished substantially by the relative wave energy or the fluctuation level with respect the background field. In addition, a set of quasilinear diffusion equations is derived for the reduced (with respect to the perpendicular velocity component) velocity distribution functions (VDFs), as they occur in the wave dispersion equation and the related dielectric function for parallel propagation. These reduced VDFs allow one to describe adequately the most prominent observed features, such as an ion beam and temperature anisotropy, in association with the resonant interactions of the particles with the waves on a kinetic level, yet have the advantage of being only dependent upon the parallel velocity component.

Electron Cyclotron Absorption by RF Driven Current-Carrying Plasmas With Fast Particle Losses

We use quasilinear theory to investigate the influence of losses of energetic particles on the behaviour of electron cyclotron absorption during lower hybrid plus electron cyclotron current drive. Our analysis takes into account the quasilinear effects of the interaction of electron cyclotron radiation with a lower hybrid produced tail in the presence of collisions and a loss term for energetic particles. We study the behaviour of different quantities, as the parallel electron distribution function and the perpendicular temperature, for several values of the electron confinement time, an important parameter of the model loss term. The absorption of extraordinary mode electron cyclotron waves by tail electrons results to be substantially reduced when the loss term is taken into account in the quasilinear diffusion equation, as compared to the case where these losses are neglected.

Some results about the approximate controllability property for quasilinear diffusion equations

We study the approximate controllability property for y t - δ( y) = uχ w, on Ω × (0, T), where Ω is a bounded open set of ℝ N and ω ⊂Ω. First, we show some negative results for the case (s) = |s| m-1s, 0 < m < 1, by means of an obstruction phenomenon. In a second part, we obtain a positive answer on the space H −1−γ (Ω), for any γ > 0, for a class of functions φ essentially linear at infinity, by using a higher order vanishing viscosity argument.

Quasilinear theory for a tokamak plasma in the presence of cyclotron resonance

Quasilinear diffusion equations for the distribution function of trapped and circulating particles interacting with waves in a tokamak by means of cyclotron resonance are derived. The resulting equations reveal new features of quasilinear diffusion and are of two kinds, one which involves bounce resonances overlapping in velocity space and one with well separated bounce resonances. These two cases correspond to situations where the phase of the wave-particle interaction between successive resonances can be considered as random or deterministic, respectively. An analysis of the conditions of applicability of the new equations is carried out and previous well known forms of the quasilinear diffusion equations are shown to be recovered in the proper limits.